Ozsvath-Szabo bordered algebras and subquotients of category O

Abstract

We show that Ozsváth-Szabó's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a q-presentable quotient of parabolic category O. We identify the endomorphism algebra of a minimal projective generator for this block with an explicit quotient of the Ozsváth-Szabó algebra using Sartori's diagrammatic formulation of the endomorphism algebra. Both of these algebras give rise to categorifications of tensor products of the vector representation V ⊗n for Uq(gl(1|1)). Our isomorphism allows us to transport a number of constructions between these two algebras, leading to a new (fully) diagrammatic reinterpretation of Sartori's algebra, new modules over Ozsváth-Szabó's algebra lifting various bases of V ⊗n , and bimodules over Ozsváth-Szabó's algebra categorifying the action of the quantum group element F and its dual on V ⊗n . Contents1. Introduction 1 2. Ozsváth-Szabó's algebras 7 3. Sartori's algebras 10 4. A surjective homomorphism from Ozsváth-Szabó to Sartori 13 5. Characterizing illicit morphisms via generating intervals 14 6. A vanishing ideal in the Sartori algebra 20 7. Fork elements and injectivity 21 8. Categorification of bases and bilinear forms 23 9. Bimodules for quantum group generators 31 References 34

Trivalent vertices and bordered knot Floer homology in the standard basis

“…Remark A.8. Due to changes of convention (apparently related to the difference between V and V * ), this identification of V ⊗n with an exterior algebra differs from the ones considered in [Mani19,LM21].…”

“…Canonical bases. We now review the canonical basis of V ⊗n used in [Sart16,LM21] (related to the basis used in [Mani19]). [Sart16].…”

“…The above three categorifications of V ⊗n are closely related to the bordered-Floer "strands algebras" A(Z) for (generalized) arc diagrams 2 Z (Sartori's categorification is more algebraic but is closely related [LM21] to Ozsváth-Szabó's bordered HFK). Specifically, the diagrams Z in question appear as three of the four vertices of a square of diagrams; see Figure 3.…”

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

“…The connection between Fukaya categories of symmetric products and convolution algebras associated to cyclic arrangements is interesting in both directions. In [LM19], the first and third authors used the algebras of Ozsváth-Szabó to construct categorical representations of gl(1|1); these constructions are a small part of a larger program of the third author and Rouquier to develop the foundations of the higher representation theory of gl(1|1) (see also [Man19], [EPV19], and [Tia16,Tia14] for closely related work). Thus, in addition to being basic ingredients to Heegaard Floer homology, the Ozsváth-Szabó algebras are also basic objects in gl(1|1) representation theory, and Theorem 1.2 implies several interesting facts about these algebras.…”

“…This corollary is useful as a tool to further understand the categorification of bases for the U q (gl(1|1)) representations initiated in [LM19]. First, the affine quasi-hereditary structure of B(V) includes as part of the structure a family of standard modules over Ozsváth-Szabó's algebras categorifying the standard tensor-product basis of V ⊗n .…”